Solving the Heat Equation for Initial Conditions

[Raven2816]

Problem:
u (sub t) = (1/2)u (sub xx)

find the solution u(x,t) of the heat equation for the following initial conditions:

u(x,0) = x
u(x,0) = x^2
u(x,0) = sinx
u(x,0) = 0 for x < 0 and 1 for x>=0

i'm really flying blind here. I've taken differential equations years ago but nothing is too familiar. i know this is second order and that's really confusing me.

so for the x^2 condition I've tried differentiating up to 3 times and simplifying. i got a solution: x^2 + t. i got it by accident so it probably isn't right.
i feel like since there are no boundaries i should be able to integrate both sides, and the plug in my initial conditions but I'm just confused in general. everything i look up online has boundaries so I'm struggling to find a comparable example to learn from.

any tips or advice would be a great help.

thanks in advance

 

[Mapes]

This problem is sometimes called "diffusion on the whole line." It's covered in most texts on partial differential equations. It's not worth it to attack it by trial and error.

 

[gabbagabbahey]

The easiest way to approach this problem is probably to use the method of "Separation of Variables". That is; assume the solution is of the form $u(x,t)=X(x)T(t)$, substitute this assumed form into your PDE and solve the resulting ODEs.